Before Paul Simon's popular song, the film was already well known as the premier color slide film (they offered Kodachrome movie film as well). It was invented by two musicians, one of whom was both a violinist and a chemist, and marketed by Kodak beginning in 1935. The famous 1985 National Geographic portrait "Afghan Girl" by Steve McCurry was taken using Kodachrome. "Nice bright colors".
Other transparency films, like AGFA (I used AGFA for a time myself) and Velvia (a Fuji product), replaced it in popularity, but not because they were necessarily "better". Rather, the newer films were preferred by publishers for producing pictures that were more eye catching to customers. As one person described it, the difference was seeing a pretty picture with Velvia vs a feeling of actually being there when viewing the same subject taken with Kodachrome.
The reason for Kodachrome's color quality has to do with its emulsion - the thin layer of chemicals which capture the light while the shutter is open. Other films have three layers of emulsion, one each for red, blue, and green and also use chemicals called "couplers" which attach the colors across the layers when exposed to light to create the right mix of RBG. Kodachrome does not have those couplers so has a thinner emulsion. It uses three layers of silver halides - like regular black and white film - with each layer suspended in a different light filtering medium. The medium makes each layer sensitive to a different color. Instead of couplers, it uses a three stage developing process done after exposing the film to add the colors, which requires a separate stage for each of the three layers, resulting in sharper pictures and brighter colors. The lack of couplers in the film also means that the film lasts longer both before and after exposure. Kodachrome slides hold their colors for a long time. I have some Kodachrome slides which my father took in the 1950s which look just as good today as they did back then. But the developing process is time consuming and tricky. It is also requires a lot of care with all the very environmentally un-friendly chemicals it uses.
When I first got into photography in the early 1960s, I used Kodachrome because color print films in those days faded fairly rapidly with age. I also used AGFA and Kodak's Ektachrome, which is still around. The latter is much easier to develop that Kodachrome - even a hobbyist can do it at home. In the 1980s I switched to print films which had much improved by then and were more convenient to share than slides, and found that Kodak films worked well for pictures in which blues and reds dominated - most California scenes for example - but Fuji films were better with greens and blues - such as Hawaii and Fiji (that's just my subjective opinion). In the last few years, I've switched over entirely to digital picture taking.
Of course, nowadays, digital photography dominates the stage. Kodak gets 70 percent of its business from digital products and Kodachrome is down to less than 1 percent of sales. Due to the complex processing it requires there is only one Kodak certified lab left in the world which handles the film (Dwayne's Photo in Kansas).
The last rolls of the film which Kodak has in stock will be given to the George Eastman House International Museum of Photography and Film in Rochester, N.Y., with one roll first being shot by Steve McCurray. I wonder what his subjects will be?